Cohomological equation and cocycle rigidity of discrete parabolic actions

We study the cohomological equation for discrete horocycle maps on $ {\rm SL}(2, \mathbb{R}) $ and $ {\rm SL}(2, \mathbb{R})\times {\rm SL}(2, \mathbb{R}) $ via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of $ {\rm SL}(2, \mathbb{R}) $. Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of $ \operatorname{\mathfrak s\mathfrak l}(2, \mathbb{R}) $, and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for $ {\rm SL}(2, \mathbb{R}) $ in which all cases of irreducible, unitary representations of $ {\rm SL}(2, \mathbb{R}) $ can be studied simultaneously.

Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.